A Pair of Multiplication-Type Operators in Quaternionic Analysis and the 2-Cauchy-Fueter Equation
Yong Li, Yuchen Zhang
TL;DR
The paper develops multiplication-type operators $L_0$ and $L_1$ to generate $k$-regular functions from $(k+1)$-regular ones, addressing lack of closure under multiplication in quaternionic analysis. It connects this construction to the $2$-Cauchy–Fueter equation by introducing a new acyclic resolution of the sheaf $\\\mathcal{R}^{(2)}$ and proving a topological solvability criterion: the equation $\\mathscr{D}^{(2)}f=g$ is solvable on a domain iff $H^3(\\Omega,\\mathbb{R})=0$, equivalently every real-valued harmonic function is the real part of a quaternionic regular function. The work also proves an inverse problem: under a harmonic-closure framework, any harmonic function admitting a representation $h=L_0(f_0,f_1)+L_1(g_0,g_1)$ can be decomposed into quaternionic-regular components, tying cohomological properties to constructive regularity. Collectively, these results link quaternionic regularity with de Rham cohomology and provide explicit mechanisms to construct regular functions from harmonic data.
Abstract
In this paper, we introduce a pair of multiplication-like operations, $L_0$ and $L_1$, which derive $k$-regular functions from $(k+1)$-regular functions. The investigation of the inverse problem naturally leads to a deeper study of the 2-Cauchy-Fueter equation. In doing so, we provide a new acyclic resolution for the sheaf of $2$-regular functions $\mathcal{R}^{(2)}$. Furthermore, a complete topological characterization for the solvability of the $2$-Cauchy-Fueter equation is established. Specifically, we prove that the $2$-Cauchy-Fueter equation $$\mathscr{D}^{(2)}f=g$$ is solvable for any $g$ satisfying $\mathscr{D}_1^{(2)}g=0$ on a domain $Ω\subset\mathbb{R}^4$ if and only if $H^3(Ω, \mathbb{R}) = 0$, or equivalently, if and only if every real-valued harmonic function on $Ω$ can be represented as the real part of a quaternionic regular function.